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  "body": "\n## Definition\n\nLet $(K, +, \\cdot, 0, 1)$ be an [[idempotent semiring|NA44T2NZE2]], with the order $a \\le b \\iff a + b = b$.\nLet $^*$ be a unary operation on $K$.\nThe structure $(K, +, \\cdot, ^*, 0, 1)$ is a *Kleene algebra* if for all $a, b, x \\in K$:\n\n1. $1 + a \\cdot a^* \\le a^*$;\n2. $1 + a^* \\cdot a \\le a^*$;\n3. if $b + a \\cdot x \\le x$, then $a^* \\cdot b \\le x$;\n4. if $b + x \\cdot a \\le x$, then $b \\cdot a^* \\le x$.\n\n## Remarks\n\nClauses 1 and 3 say that $a^* b$ is the least $x$ with $b + a x \\le x$.\nExample: let $\\Sigma^*$ be the [[free monoid|27ZPTJP9PP]] on an alphabet $\\Sigma$, and let $K$ be the set of languages $L \\subseteq \\Sigma^*$.\nWith union, concatenation, $0 = \\emptyset$, $1 = \\set{\\varepsilon}$ and $L^* = \\bigcup_{n \\ge 0} L^n$, this is a Kleene algebra.\nIndeed, $L^* B$ is the set of words $w_1 \\cdots w_n u$ with $w_i \\in L$ and $u \\in B$, and each such word lies in any $X$ with $B \\cup L X \\subseteq X$, by induction on $n$.\nExample: the [[binary relations|08EJC6Q25H]] on a set, with union, composition and reflexive transitive closure as $^*$, form a Kleene algebra.\nNon-example: the Boolean semiring $\\set{0, 1}$ with $a^* = a$ fails clause 1, since for $a = 0$ it says $1 + 0 \\le 0$.\nWith $a^* = 1$ for all $a$, the Boolean semiring is a Kleene algebra.\nIn every Kleene algebra $a^* = 1 + a a^*$ and $a^* a^* = a^*$.\n",
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